Quantum_ Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar [60]
In 1892, improved equipment appeared to show that the red alpha and blue gamma Balmer lines of the hydrogen spectrum were not single lines at all, but were each split in two. For more than twenty years, it remained an open question whether these lines were 'true doublets' or not. Bohr thought not. It was at the beginning of 1915 that he changed his mind as new experiments revealed that the red, blue and violet Balmer lines were all doublets. Using his atomic model, Bohr could not explain this 'fine structure', as the splitting of the lines was called. As he settled into his new role as a professor in Copenhagen, Bohr found a batch of papers waiting for him from a German who had solved the problem by modifying his atom.
Arnold Sommerfeld was a 48-year-old distinguished professor of theoretical physics at Munich University. Over the years, some of the most brilliant young physicists and students would work under his watchful eye as he turned Munich into a thriving centre of theoretical physics. Like Bohr, he loved skiing and would invite students and colleagues to his house in the Bavarian Alps to ski and talk physics. 'But let me assure you that if I were in Munich and had the time, I would sit in on your lectures in order to perfect my knowledge of mathematical physics', Einstein had written to Sommerfeld in 1908 while still at the Patent Office.48 It was some compliment coming from a man described as a 'lazy dog' by his maths professor in Zurich.
To simplify his model, Bohr had confined electrons to move only in circular orbits around the nucleus. Sommerfeld decided to lift this restriction, allowing electrons to move in elliptical orbits, like the planets in their journey around the sun. He knew that, mathematically speaking, circles were just a special class of ellipse, therefore circular electron orbits were only a subset of all possible quantised elliptical orbits. The quantum number n in the Bohr model specified a stationary state, a permitted circular electron orbit, and the corresponding energy level. The value of n also determined the radius of a given circular orbit. However, two numbers are required to encode the shape of an ellipse. Sommerfeld therefore introduced k, the 'orbital' quantum number, to quantise the shape of an elliptical orbit. Of all the possible shapes of an elliptical orbit, k determined those that were allowed for a given value of n.
In Sommerfeld's modified model, the principal quantum number n determined the values that k could have.49 If n=1, then k=1; when n=2, k=1 and 2; when n=3, k=1, 2 and 3. For a given n, k is equal to every whole number from 1 up to and including the value of n. When n=k, the orbit is always circular. However, if k is less than n, then the orbit is elliptical. For example, when n=1 and k=1, the orbit is circular with a radius r, called the Bohr radius. When n=2 and k=1, the orbit is elliptical; but n=2 and k=2 is a circular orbit with a radius 4r. Thus, when the hydrogen atom is in the n=2 quantum state, its single electron can be in either the k=1 or k=2 orbits. In the n=3 state, the electron can occupy any one of three orbits: n=3 and k=1, elliptical; n=3, k=2, elliptical; n=3 and k=3, circular. Whereas in Bohr's model n=3 was just one circular orbit, in Sommerfeld's modified quantum atom there were three permitted orbits. These extra stationary states could explain the splitting of the spectral lines of the Balmer series.
Figure 8: Electron orbits for n=3 and k=1, 2, 3 in the Bohr-Sommerfeld model of the hydrogen atom
To account for the splitting of the spectral lines, Sommerfeld turned to Einstein's theory of relativity. Like a comet in orbit about the sun, as an electron in an elliptical orbit heads towards the nucleus its speed increases. Unlike a comet, the speed of the electron is great enough for its mass to increase